Problem Bank

Positive integers $x, y, z$ satisfy

Find the maximum of $x^2 + y^2 + z^2$.

Let $f(x) = \ln x - ax$ and $g(x) = \mathrm{e}^x - ax$, where $a$ is a real number.

1. If $f(x)$ is decreasing on $(1, +\infty)$ and $g(x)$ has a minimum value on $(1, +\infty)$, …

The function $f$ is odd, defined on all of $\mathbb{R}$, has period $3$, and

How many sol…

A regular tetrahedron $ABCD$ has edge length $2\sqrt{2}$, and a point $P$ satisfies

Find the minimum and maximum v…

The circle $O:x^2+y^2=1$ is the circumscribed circle of $\triangle ABC$, and $\tan A=2$. If $\overrightarrow{AO}=x\overrightarrow{AB}+y\overrightarrow{AC}$, find the maximum value …

Through the focus $F$ of the parabola $E \colon x^2 = 2py$ (with $p > 0$), two distinct lines $l_1$ and $l_2$ are drawn with slopes $k_1$ and $k_2$ satisfying $k_1 + k_2 = 2$. Line…

Let $f(x) = \ln x - \dfrac{a}{2}x^2 + x + 2$ with $a \in \mathbb{R}$.

1. Discuss the monotonicity of $f(x)$. 2. Let $g(x) = f(x) + bx - 2$ with $a \neq 0$, and suppose $g(x)$ has …

A fraction $\dfrac pq$ (with $p,q$ positive integers) has decimal expansion $\dfrac pq=0.198\ldots$ (that is, its decimal begins $0.198$). When $q$ is as small as possible, find $p…

In the isosceles right triangle $OPQ$, $\angle POQ = 90^\circ$ and $OP = 2\sqrt{2}$. A point $M$ lies on the segm…

Consider the parabola $x^2=4y$. A point $A$ on the parabola lies in the first quadrant, and the tangent line $l$ at $A$ meets the $x$-axis at a point $B$. The line $l'$ through $B$…

Let $n$ be a given positive integer, and let $a_1, a_2, \dots, a_n$ be real numbers such that for every $m \leqslant n$,

Find the remainder when

is divided by $x^2 + 1$.